Abstract:
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finite-dimensional representation. In one-dimensional case a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$) and $gl_2 ({\bold R})_K$ ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented.

Abstract:
It is shown that the Coulomb correlation problem for a system of two electrons (two charged particles) in an external oscillator potential possesses a hidden $sl_2$-algebraic structure being one of recently-discovered quasi-exactly-solvable problems. The origin of existing exact solutions to this problem, recently discovered by several authors, is explained. A degeneracy of energies in electron-electron and electron-positron correlation problems is found. It manifests the first appearence of hidden $sl_2$-algebraic structure in atomic physics.

Abstract:
A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.

Abstract:
Polynomial relations between the generators of $q$--deformed Heisenberg algebra invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the following: if two elements $a$ and $b$, obeying the relation \[ ab - q ba = p, \] where $p, q$ are any complex numbers, then for any $p,q$ and natural $n$ \[ (aba)^n = a^n b^n a^n \]

Abstract:
A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra $osp(2,2)$ in the "projectivized" representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

Abstract:
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial element of the universal enveloping algebra of $sl_2({\bf R})$ (for differential equations) or $sl_2({\bf R})_q$ (for finite-difference equations) in the "projectivized" representation possessing an invariant subspace. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

Abstract:
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the eigenvalue problem for a polynomial elements of the universal enveloping algebras of the algebras $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$ and $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$ taken in the "projectivized" representations (in differential operators of the first order in two real variables) possessing an invariant subspace. General insight to the problem of a description of linear differential operators possessing an invariant sub-space with a basis in polynomials is presented. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.

Abstract:
We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratios of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves.

Abstract:
Two Lie algebraic forms of the 2-body Elliptic Calogero model are presented. Translation-invariant and dilatation-invariant discretizations of the model are obtained.